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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Maximal complexifications of certain homogeneous Riemannian manifolds


Authors: S. Halverscheid and A. Iannuzzi
Journal: Trans. Amer. Math. Soc. 355 (2003), 4581-4594
MSC (2000): Primary 32C09, 53C30, 32M05; Secondary 32Q28
Published electronically: July 8, 2003
MathSciNet review: 1990763
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Abstract: Let $M=G/K$ be a homogeneous Riemannian manifold with $\dim_{\mathbb{C}} G^{\mathbb{C}} = \dim_{\mathbb{R}} G$, where $G^{\mathbb{C}}$ denotes the universal complexification of $G$. Under certain extensibility assumptions on the geodesic flow of $M$, we give a characterization of the maximal domain of definition in $TM$ for the adapted complex structure and show that it is unique. For instance, this can be done for generalized Heisenberg groups and naturally reductive homogeneous Riemannian spaces. As an application it is shown that the case of generalized Heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex.


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Additional Information

S. Halverscheid
Affiliation: Carl von Ossietzky Universität, Fachbereich Mathematik, D-26111 Oldenburg i. O., Germany
Email: halverscheid@mathematik.uni-oldenburg.de

A. Iannuzzi
Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, I-40126 Bologna, Italy
Email: iannuzzi@dm.unibo.it

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03263-X
PII: S 0002-9947(03)03263-X
Keywords: Adapted complex structure, homogeneous Riemannian spaces, Stein manifold
Received by editor(s): December 31, 2001
Received by editor(s) in revised form: November 18, 2002
Published electronically: July 8, 2003
Additional Notes: The first author was partially supported by SFB 237, “Unordnung und große Fluktuationen"
The second author was partially supported by University of Bologna, funds for selected research topics
Article copyright: © Copyright 2003 American Mathematical Society